
denote by \(\iota :U_0\rightarrow U,\jmath :V_0\rightarrow V,q:(U\times V)_0\rightarrow U\times V\)  the closed embeddings of the zero set of \(f,g,f\boxplus g\) . One has the closed embedding \(\tilde{U}_0\times \tilde{V}_0\rightarrow (\tilde{U\times V})_0\) , and [1]} shows that the natural arrow of the six-functor formalism \(R\Gamma _{\tilde{U}_0\times \tilde{V}_0}\rightarrow R\Gamma _{(\tilde{U\times V})_0}\)  is an isomorphism. The Thom-Sebastiani isomorphism is then defined by the composition of natural isomorphisms of the six-functor formalism:
\(\phi _f\boxtimes \phi _g=&\iota ^\ast R\Gamma _{\tilde{U}_0}\boxtimes \jmath ^\ast R\Gamma _{\tilde{V}_0}\nonumber \\\simeq &(\iota \times \jmath )^\ast R\Gamma _{\tilde{U}_0\times \tilde{V}_0}\nonumber \\\overset{\simeq }{\rightarrow }& k^\ast q^\ast R\Gamma _{\tilde{U\times V}_0}\nonumber \\=&k^\ast \phi _{f\boxplus g}\) 
